Geology Reference
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1.3.5 Example 1-7. Problems without initial conditions.
Very often, vibrations may have been maintained over large enough time
scales, that the effects of attenuation have rendered the influence of initial
conditions insignificant. Thus we consider “problems without initial
conditions,” or “dynamically steady wave motions” where oscillations at
constant frequency are maintained. These formulations are useful because the
required labor is reduced, and the results, unlike series expansions, are amenable
to physical interpretation. Because steady state formulations lead to usable
results more rapidly than general solutions, we can increase the complexity of
the underlying physical model. Equation 1-1, for example, describes undamped
motions only, and arises as an approximation to the dissipative formulation
2
u/ t
2
+
u/ t - c
2
2
u/ x
2
= 0
(1.35)
We will solve both Equations 1.1 and 1.35, using dynamically steady
methods and introduce simple concepts from complex variables (Equation 1.35
is discussed later in a vibrations context). Again, the undamped equation is
2
u/ t
2
- c
2 2
u/ x
2
= 0 (1.36)
We consider 0 x , and prescribe the end displacement u(0,t) = A cos t,
where and the amplitude A are real. The resulting waves clearly propagate to
the right. We solve this problem using several elementary methods,
demonstrating the advantages and disadvantages of each.
1.3.5.1 Example 1-7a. Naïve approach.
Students often wonder why complex exponentials are used at all. Since
practical solutions must ultimately appear in real terms, why should imaginary
numbers be used in the solution process? We attempt to solve the problem using
real functions only, pretending that complex functions do not exist. Knowing
that trigonometric functions yield simplifications, we assume the separable
solution for a
real
function u(x,t) as
u(x,t) A cos t U(x) (1.37)
where the real function U(x) is to be determined. The cos t in Equation 1.37 is
motivated by its appearance in the boundary condition u(0,t) = A cos t. If we
substitute Equation 1.37 into Equation 1.36, we obtain the one-dimensional
Helmholtz equation
U"(x) +
2
/c
2
U(x) = 0
(1.38)
whose solution is U(x) = C
1
sin
x/c + C
2
cos
x/c. We may write u(x,t) C
3
cos
t sin
x/c + C
4
cos
t cos
x/c, on replacing AC
1
and AC
2
by C
3
and C
4
.
The identities cos
t sin
x/c = 1/2 {sin
(x/c - t) + sin
(x/c + t)} and
cos
t cos
x/c = 1/2 {cos
(x/c - t) + cos
(x/c + t)} allow us to recast our
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